Mannheimer and Schechter model revisited: Viscosimetry of a (non-)Newtonian and curved interface

The impact of a Newtonian or non-Newtonian interface, possibly curved due to wetting effects, on the surface viscosimetry method historically developed by Mannheimer and Schechter [Mannheimer and Schechter, J. Colloid Interface Sci. 32, 195 (1970); Mannheimer and Schechter, J. Colloid Interface Sci. 32, 212 (1970)] is investigated theoretically and numerically in this paper. The Reynolds number is considered small enough for inertial effects to be negligible and thus for surface shear viscosity to be preferentially studied. The classical Boussinesq-Scriven law is modified by a shear-dependent viscosity governed by the Carreau constitutive law. The case of a curved interface is also investigated whether the liquid is hydrophilic or hydrophobic (liquid metals), which is revealed to be relevant for a liquid with a significant capillary length. The shallow-channel layout is found to be particularly well suited to the detection of a non-Newtonian liquid surface whatever its mechanical behavior: shear thinning or shear thickening. A key to evaluating the parameters involved in a surface viscosity model is to carry out a series of experiments at various rotation speeds, with preferably a nonwetting contact angle. Indeed, it is demonstrated that such a contact angle tends to amplify non-Newtonian effects, especially for a carrier fluid with a significant capillary length.

Figure 1

Cross section of the annular surface viscometer considering the shallow-channel configuration on the left-hand side and the deep-channel layout on the right-hand side.

Figure 2

Three-dimensional representation of the liquid flowing in the annular viscometer with an example of curved interface (inner radius $r_i=3$ cm, outer radius $r_o=7$ cm, and average liquid depth $h_0=1$ cm).

Figure 3

Geometry and boundary conditions of the 2D axisymmetric model.

Figure 4

(a) Surface velocity profile, (b) surface shear, and (c) bulk shear at the interface obtained by numerical simulations and compared to the analytical solution of Mannheimer and Schechter for various Boussinesq numbers ($n=1$ and $\varphi ={90}^{\circ }$).

Figure 5

(a) Surface velocity profile, (b) Boussinesq number along the interface, (c) surface shear, and (d) bulk shear at the interface for the Newtonian (MS), shear-thinning ($n=0.5$), and shear-thickening ($n=2$) cases for various (Newtonian) Boussinesq numbers ($\text{Re}=100$, $\lambda =100$, and $\varphi ={90}^{\circ }$).

Figure 6

(a) Surface velocity profile and (b) Boussinesq number along the interface for the Newtonian (black solid curve), shear-thinning ($n=0.5$), and shear-thickening ($n=2$) cases for various relaxation times $\lambda$ ($\text{Re}=100$, $\text{Bo}=0.05$, and $\varphi ={90}^{\circ }$).

Figure 7

Evolution of the bath height for an aqueous solution ($l_c=2$ mm) and molten aluminium ($l_c=6$ mm) for various contact angles.

Figure 8

(a) and (b) Surface velocity profile and (c) and (d) surface shear along a Newtonian interface ($n=1$), for the wetting ($\varphi ={30}^{\circ }$), planar (MS), and nonwetting ($\varphi ={150}^{\circ }$) cases, as a function of the Boussinesq number and the liquid bath: (a) an aqueous solution ($l_c=2$ mm) or (b)–(d) molten aluminium ($l_c=6$ mm).

Figure 9

Bulk velocity field for (a) the wetting case ($\varphi ={30}^{\circ }$) and (b) the nonwetting case ($\varphi ={150}^{\circ }$), considering the molten aluminium subphase ($l_c=6$ mm) with $\text{Bo}=0.001$.

Figure 10

Surface velocity profile for (a) a shear-thinning ($n=0.5$) and (b) a shear-thickening ($n=2$) interface for various Boussinesq number and contact angle, considering the molten aluminium subphase ($l_c=6$ mm, $\lambda =100$, and $\text{Re}=100$).

Figure 11

(a) Maximum surface velocity along the interface for the shear-thinning ($n=0.5$), Newtonian ($n=1$), and shear-thickening ($n=2$) cases as a function of the Boussinesq number. (b) Relative difference between the Newtonian and the non-Newtonian cases. Here $\varphi ={90}^{\circ }$, $\lambda =100$, and $\text{Re}=100$.

Figure 12

Isovalues of the relative difference $E(\%)$, as a function of the Boussinesq number and the power-law index $n$ for (a) $\lambda =200$ and (b) $\lambda =2000$, with $\varphi ={90}^{\circ }$ and $\text{Re}=200$.

Figure 13

Relative difference of the maximum surface velocities between the Newtonian and the non-Newtonian cases, with (a) a shear-thickening interface ($n=0.5$) and (b) a shear-thinning interface ($n=2$), for various contact angles considering molten aluminium ($l_c=6$ mm, $\text{Re}=100$, and $\lambda =100$).

Figure 14

(a) Dependence of the surface velocity profiles on the Reynolds number for a shear-thinning ($n=0.5$) and a shear-thickening ($n=2$) interface, with $\text{Bo}=0.05$ and $\lambda =250$. (b) Dependence of the maximum surface velocity on the Reynolds number, with $\text{Bo}=0.05$.